Why is energy quadratic in velocity?

[cesiumfrog]

(Reviving an old thread based on a recent request by "mr.smartass#1")

A sophisticated answer has to do with the notion that symmetries give rise to conserved quantities. The mathematical expression of this relationship is named Noether's theorem. In this case, the fact that the laws of physics are the same, from one day to another, necessarily implies conservation of a quantity which just turns out to be quadratic in velocity.

This is "deep" in the sense that Noether's theorem works in everything from general relativity to quantum field theory. The downside is that it needs you to start by supplying the Lagrangian. For somebody unfamiliar with this, it is going to be difficult to motivate the particular form of the Lagrangian (we can't just say T-V as that would be circular). You could derive it from Newton's laws (by following a proof of the Euler-Lagrange equations) but this seems excessively convoluted.

I think the simplest answer is just, it is a mathematical fact that $\Delta (\frac 1 2 m\,v^2) = ma\, \Delta x$. This equation simply says that the change-in-KE (the left hand side) is equal to the Work (the name given to the right hand side, which can also be written as $F\ \Delta x$ because of F=ma, so the work just measures how forcefully and how far you push). This equation is usually known in the familiar form "v²-u²=2as". The proof is basic calculus: $\frac {\delta (\frac 1 2 m\,{\dot x}^2)} {\delta x} = m \frac {\delta (\frac 1 2 {\dot x}^2)} {\delta \dot x} \frac {\delta \dot x} {\delta t} \frac {\delta t} {\delta x} = m\ \dot x \ddot x \frac 1 {\dot x}$ (using the chain rule). If there are no external forces doing work on the system then, because the internal forces are balanced (by Newton's 3rd law), this equation tells us that any internal redistributions of v² must also stay balanced (so that the total is conserved). Something like that?

 

[zoki85]

Quadratic dependence of energy on velocity is true in Newtonian mechanics, but not true in a relatvistic mechanics

 

[nottoosmart]

(Reviving an old thread based on a recent request by "mr.smartass#1")

A sophisticated answer has to do with the notion that symmetries give rise to conserved quantities. The mathematical expression of this relationship is named Noether's theorem. In this case, the fact that the laws of physics are the same, from one day to another, necessarily implies conservation of a quantity which just turns out to be quadratic in velocity.

This is "deep" in the sense that Noether's theorem works in everything from general relativity to quantum field theory. The downside is that it needs you to start by supplying the Lagrangian. For somebody unfamiliar with this, it is going to be difficult to motivate the particular form of the Lagrangian (we can't just say T-V as that would be circular). You could derive it from Newton's laws (by following a proof of the Euler-Lagrange equations) but this seems excessively convoluted.

I think the simplest answer is just, it is a mathematical fact that $\Delta (\frac 1 2 m\,v^2) = ma\, \Delta x$. This equation simply says that the change-in-KE (the left hand side) is equal to the Work (the name given to the right hand side, which can also be written as $F\ \Delta x$ because of F=ma, so the work just measures how forcefully and how far you push). This equation is usually known in the familiar form "v²-u²=2as". The proof is basic calculus: $\frac {\delta (\frac 1 2 m\,{\dot x}^2)} {\delta x} = m \frac {\delta (\frac 1 2 {\dot x}^2)} {\delta \dot x} \frac {\delta \dot x} {\delta t} \frac {\delta t} {\delta x} = m\ \dot x \ddot x \frac 1 {\dot x}$ (using the chain rule). If there are no external forces doing work on the system then, because the internal forces are balanced (by Newton's 3rd law), this equation tells us that any internal redistributions of v² must also stay balanced (so that the total is conserved). Something like that?

thank you cesiumfrog

 

[anorlunda]

A different way to look at it is that even powered terms are always positive, (1*1=-1*-1). Square is the simplest even power.

I can use energy to accelerate an object. The energy used (think fuel in a machine) has no sense of direction. The object moving in air will slow because of friction, turning the kinetic energy into heat. Heat has no direction. It would make no sense to start with scalar energy, end with scalar energy, yet have a vector energy as an intermediate step.

A moving massive object has both momentum and energy in some reference frames. Two conservation laws apply, one vector, one scalar. Isn't it unavoidable that both odd powered terms and even powered terms are needed to describe it?

 

[vanhees71]

You find the expressions for the Hamilton function (the Hamiltonian formulation of classical mechanics is much more suited to symmetry analyses than the Lagrangian one, because it provides you with the Poisson-bracket formalism which is a representation of the Lie algebra of the symmetry group of spacetime on phase space) by analyzing the Lie algebra of the Galilei group in terms of Poisson brackets. This leads you to the typical possible Hamiltonian for a closed system of many point particles,
$$H=\sum_{j=1}^{n} \frac{\vec{p}_j^2}{2m} + \frac{1}{2} \sum_{i \neq j} V(|\vec{x}_i-\vec{x}_j|).$$

 

[voko]

In my graduate course of classical mechanics, dynamics started with Newton's laws (using vectors). In the second or third lecture, a dot product of Newton's second law with velocity was taken and integrated, which introduced kinetic energy and work. Then a calculus theorem was referenced (known to us by then) on path independence of certain integrals, and potential energy was introduced formally, along with conservation of total energy.

This may not be the most profound way to introduce energy, but it was certainly very simple to understand.

 

[vanhees71]

Sure, that's the usual way to introduce the concept of energy, and it's very adequate, but do you understand from that, why there are the conservation laws of Newtonian mechanics (energy, momentum, angular momentum, center of mass, total mass)?

The concept of symmetries is a rather new concept, introduced by Einstein in his famous paper on "The Electrodynamics of Moving Bodies" (1905). Here we find in the introductory paragraph one of the first explicit symmetry arguments in the history of physics: There are "asymmetries" in the contemporary interpretations of Maxwell's theory which seem to be not present in nature, namely the induction of an electromotive force when a conductor is moving near a magnet or when the conductor is at rest while the magnet is moving. The resolution of this problem was the development of Special Relativity Theory.

10 years later Einstein developed his General Relativity Theory to describe gravitation in a way consistent with relativity, and it was troublesome to understand conservation laws, particularly the energy-conservation law, within this theory. The mathematical analysis of this problem is due to Emmy Noether who in 1918 published an article about the relation of symmetry principles and conservation laws, according to which any symmetry of the physical laws imply the existence of conserved quantities and vice versa, i.e., any conserved quantity implies a symmetry of the dynamical laws of the unerlying theory.

For classical mechanics and field theory this is just an aesthetical mathematical theorem, but it's of utmost importance in quantum theory, because it's the most important basis for the understanding of the description of observables, i.e., the operator algebra a quantum theory is determined by the symmetries of the system under consideration. For classical (Newtonian) mechanics, that implies first of all the conservation laws resulting from the Galilei-Newtonian space-time model: energy (time translation invariance), momentum (spatial translation invariance), angular momentum (isotropy of space; symmetry under rotations), center-of-mass velocity (Galilei boost invariance). The conservation of mass in Newtonian physics is more complicated, because it's fully understood only using group theory in connection with quantum theory: Mass is a socalled non-trivial central charge of the Galilei group's Lie algebra, and this implies a superselection rule, according to which in non-relativistic quantum theory one must not superimpose quantum states of systems with different mass, i.e., each system lives in its superlection sector characterized by the total mass.

 

[voko]

do you understand from that, why

"Why" is a tricky question in physics. At some stage the answer to it becomes "it is so postulated" and, if you are lucky, "it is an experimental fact". And if one is not happy with that, it becomes a philosophical question (which is verboten here, I understand). I think I am repeating the intro in #1, though.